Entry 2.1 is based on the COBE measurement of the temperature of the
thermal cosmic electromagnetic background radiation (the CMBR),
To = 2.725 K
(Mather et al. 1999).
The COBE and UBC measurements
(Mather et al. 1990;
Gush, Halpern, &
Wishnow 1990)
show that the spectrum is very close to thermal. It
has been slightly disturbed by the observed
interaction with the hot plasma in clusters of galaxies
(LaRoque et al. 2003
and references therein). The limit on the resulting fractional increase
in the radiation energy density is
(Fixsen et al. 1996)

(14)

This means that the background radiation energy density has been
perturbed by the amount
<
10-8.5. Improvements of this number are under discussion (e.g.
Zhang, Pen & Trac
2004),
and might be entered in a future version of the inventory.

The thermal background radiation has been
perturbed also by the dissipation of the primeval fluctuations in the
distributions of baryons and radiation on scales smaller than the
Hubble length at the epoch of decoupling of baryonic matter and
radiation. If the initial mass fluctuations are adiabatic and
scale-invariant the fractional perturbation to the radiation energy per
logarithmic increment of the comoving length scale is
u / u ~
h2,
where h ~
10-5 is the density
contrast appearing at the Hubble length. This is small compared to the
subsequent perturbation by hot plasma (eq [14]).

where the neutrino mass eigenstates are ordered as
m1
<
m2
<
m3.
Entry 2.2, the density parameter
in primeval neutrinos,
assumes that
me
may be neglected. The upper limit from WMAP and SDSS is
< 0.04
(Tegmark et al. 2004a).
At this limit the three families would have almost equal masses,
m = 0.6 eV.
This may not be very likely, but one certainly must bear in mind the
possibility that our entry is a considerable underestimate.

Light elements are produced as the universe expanded and cooled
through kT ~ 0.1 MeV, in amounts that depend
on the baryon abundance. The general agreement of the baryon
abundance inferred in this way with that derived from the CMBR
temperature anisotropy gives confidence that
the total amount of baryons - excluding what might have been trapped
in the dark matter prior to light element nucleosynthesis - is
securely constrained.

close to the mean of the first two. Since the relation between the
helium abundance and the baryon density parameter has a very shallow
slope, an accurate abundance estimate (say, with < 1% error) is
needed for a strong constraint on
bh2. We consider
that the current estimates may still suffer from systematic errors which
are not included in the error estimates in the literature.
3
Within the standard cosmology our adopted value in
equation (16) requires that the primeval helium abundance is

(17)

and the ratio of the total matter density to the baryon component is

(18)

We need in later sections the stellar helium production rate
with respect to that of the heavy elements. The all-sample analysis of
Izotov & Thuan
(2004)
gives Y /
Z 2.8± 0.5. The
value derived by
Peimbert, Peimbert &
Ruiz (2000)
corresponds to 2.3± 0.6.
These value may be compared to estimates from
the perturbative effects on the effective temperature-luminosity
relation for the atmosphere of main sequence dwarfs,
Y /
Z 3± 2
(Pagel & Portinari
1998),
and 2.1± 0.4
(Jimenez et al. 2003).
From the initial elemental abundance estimate in the standard solar
model of Bahcall, Pinsonneault & Basu
(2001;
hereinafter BP2000) we derive
Y /
Z 1.4.
We adopt

(19)

Nuclear binding energy was released during nucleosynthesis. This
appears in entry 2.3 as a negative value, meaning the comoving
baryon mass density has been reduced and the energy density in
radiation and neutrinos increased. The effect on the radiation
background has long since been thermalized, of course, but the entry is
worth recording for comparison to the nuclear binding energy released
in stellar evolution. For the same reason, we compute the binding
energy relative to free protons and electrons. The convention is
artificial, because light element formation at high redshifts was
dominated by radiative exchanges of neutrons, protons and atomic
nuclei, and the abundance of the neutrons was determined by energy
exchanges with the cosmic neutrino background. It facilitates
comparison with category 6, however.
The nuclear binding energy in entry 2.3 is the product

(20)

This is larger in magnitude than the energy in the CMBR today.

3 We note, as an indication of the
difficulty of these observations, that helium abundances
inferred from the triplet 4d-2p transition
(4471) are lower than
what is indicated by the triplet 3d-2p
(5876) and singlet
3d-2p (6678)
transitions, by an amount that is significantly larger than
the quoted errors. Another uncertainty arises from
stellar absorption corrections, which are calculated only for
the 4471 line. The
table given in Izotov and Thuan
suggests that a small change in absorption corrections for
the 4471 line induces a
sizable change in the final
helium abundance estimate. We must remember also that
Y /
Z is not
very well determined.
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